A088537 Decimal expansion of Madelung's constant M2.

1, 6, 1, 5, 5, 4, 2, 6, 2, 6, 7, 1, 2, 8, 2, 4, 7, 2, 3, 8, 6, 7, 9, 2, 3, 3, 3, 2, 7, 5, 8, 6, 1, 8, 0, 9, 0, 1, 9, 6, 4, 2, 2, 9, 2, 3, 6, 1, 3, 7, 7, 7, 1, 4, 5, 6, 9, 3, 7, 3, 5, 3, 5, 9, 6, 1, 2, 6, 5, 1, 2, 3, 1, 6, 1, 5, 3, 3, 3, 6, 2, 9, 0, 4, 1, 6, 5, 8, 9, 5, 5, 1, 7, 1, 8, 7, 2, 1, 4, 5, 5, 7, 4, 9, 0

Offset

1,2

References

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 76-81.

Links

Table of n, a(n) for n=1..105.

I. J. Zucker, Exact results for some lattice sums in 2, 4, 6 and 8 dimensions, J. Phys. A: Math., Gen. vol. 7 (1974) no. 13, pp. 1568-1575.

Formula

M2 = Sum_{ -oo < i < oo, -oo < j < oo, (i,j) != (0,0) } (-1)^(i + j)/sqrt(i^2 + j^2)).

M2 = 4*(sqrt(2) - 1)*zeta(1/2)*beta(1/2) (beta=Dirichlet beta function).

Example

M2 = -1.6155426267....

Maple

M2:=evalf(4*(sqrt(2)-1)*Zeta(1/2)*sum('(-1)^n/sqrt(2*n+1)', 'n'=0..infinity), 120); # Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 10 2009

Mathematica

(2-2*I)*(Sqrt[2]-1)*Zeta[1/2]*(PolyLog[1/2, -I]-Zeta[1/2, 1/4]) // Re // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Feb 15 2013 *)

Prog

(PARI) DirBet=sumalt(n=0, (-1)^n/sqrt(2*n+1)); print(4.0*(sqrt(2)-1)*zeta(0.5)*DirBet) ; \\ R. J. Mathar, Jul 20 2007

Crossrefs

Cf. A059750.

Sequence in context: A021623 A197296 A177838 * A372065 A370424 A325313

Adjacent sequences: A088534 A088535 A088536 * A088538 A088539 A088540

Keyword

nonn,cons

Author

Benoit Cloitre, Nov 16 2003

Extensions

More terms from R. J. Mathar, Jul 20 2007

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 10 2009

Status

approved